Internal combustion engine torque optimization subject to constraints in hybrid-electric powertrains

ABSTRACT

Hybrid-electric powertrain control systems and methods include a set of sensors configured to measure a set of operating parameters of a set of components of the hybrid-electric powertrain that are each a constraint on minimum/maximum motor torque limits for the electric motors thereby collectively forming a set of constraints and a controller configured to perform a linear optimization to find the best engine torque values that are attainable subject to a set of constraint inequalities as defined by current vehicle state and as monitored by the set of sensors, and control the hybrid-electric powertrain based on the best engine torque values to avoid excessive torque commands that could damage physical components of the hybrid-electric powertrain and/or could cause undesirable noise/vibration/harshness (NVH) characteristics.

FIELD

The present application generally relates to vehicles havinghybrid-electric powertrains and, more particularly, to internalcombustion engine torque optimization subject to constraints inhybrid-electric powertrains.

BACKGROUND

A vehicle hybrid-electric powertrain typically includes an internalcombustion engine and at least one electric motor. In someconfigurations, a hybrid-electric powertrain could include an engine andtwo or more electric motors. Various components of the powertrain, e.g.,clutches, drive belts, electric motors, etc. offer a limited capacity totransmit torque. A physics-based analysis determines the inequalitiesthat link the minimum and maximum torque limits of mechanical componentsto the motor and engine torque levels. If the physics-based analysisshows that a constrained variable (e.g., clutch torque) depends on thetorque levels of both electric motors, then this is referred to as“sloped constraints,” with the horizontal constraints only depending onthe torque level of one of the electric motors and vertical constraintsonly depending on the torque level of the other electric motor. Incontrast, if the constrained variable were not sloped, it would dependon only one electric motor torque level, and thus there would be fewervariables to solve for when the constraints and minimum and maximumvalues are calculated. The challenge of working with sloped constraintsis that they lead to more complex constraint equations to solve. Some ofthese equations, or even systems of equations, are so complex, that avehicle controller could lack the processor execution speed and memoryto handle them with the required update frequency. Accordingly, whilesuch conventional hybrid-electric powertrain control systems do work fortheir intended purpose, there exists an opportunity for improvement inthe relevant art.

SUMMARY

According to one example aspect of the present invention, a controlsystem for a hybrid-electric powertrain of a vehicle, thehybrid-electric powertrain including an internal combustion engine, twoelectric motors, a battery system, and an optional transmissioncomprising a plurality of clutches, is presented. In one exemplaryimplementation, the control system comprises a set of sensors configuredto measure a set of operating parameters of a set of components of thehybrid-electric powertrain, wherein each component of the set ofcomponents is a constraint on minimum/maximum motor torque limits forthe electric motors thereby collectively forming a set of constraints,and a controller configured to perform a linear optimization to find thebest engine torque values that are attainable subject to a set ofconstraint inequalities as defined by current vehicle state and asmonitored by the set of sensors, and control the hybrid-electricpowertrain based on the best engine torque values to avoid excessivetorque commands that could damage physical components of thehybrid-electric powertrain and/or could cause undesirablenoise/vibration/harshness (NVH) characteristics.

In some implementations, there is a plurality of constraints dependingon a configuration of the hybrid-electric powertrain and operation mode.In some implementations, the plurality of constraints includes at leasttwo of engine torque, drive belt slip torque, torque of the electricmotors, clutch and/or transmission gear states, and a state of thebattery system. In some implementations, the operation mode includes atleast one of whether the engine participates in a series hybridoperating mode or a parallel hybrid operating mode, electric motorgenerator or motor mode, battery system depletion, charge sustaining, orrecharging modes, clutch pack slippage, disengaged/open, or lockoperation, dog clutch engaged or disengaged operation, and transmissiongear selection.

In some implementations, the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations, and then the best enginetorque levels are chosen. In some implementations, the controller firstdetermines four intersection lines between the two pairs of constraintplanes and then finds the intersection points of these lines with theplanes of the motor minimum/maximum torque levels with athree-dimensional geometric approach.

In some implementations, the controller determines the intersectionpoints of two of the constraint plane pairs with the slab, and then thebest engine torque levels are chosen. In some implementations, thecontroller is configured to first find the four vertical edges of themotor minimum/maximum torque slab and then find the intersection pointsof two constraint plane pairs at a time with the four vertical edgesleading to at most sixteen intersection points.

In some implementations, the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations and determine the intersectionpoints of two of the constraint plane pairs with the slab, and then thebest engine torque levels are chosen. In some implementations, thecontroller determines four intersection lines between the two pairs ofconstraint planes and then finds the intersection points of these lineswith the planes of the motor minimum/maximum torque levels with athree-dimensional geometric approach, and finds the four vertical edgesof the motor minimum/maximum torque slab and then finds the intersectionpoints of two constraint plane pairs at a time with the four verticaledges leading to at most sixteen intersection points.

According to another example aspect of the invention, a method forcontrolling for a hybrid-electric powertrain of a vehicle, thehybrid-electric powertrain including an internal combustion engine, twoelectric motors, a battery system, and an optional transmissioncomprising a plurality of clutches, is presented. In one exemplaryimplementation, the method comprises receiving, by a controller of thevehicle and from a set of sensors of the hybrid-electric powertrain, aset of operating parameters of a set of components of thehybrid-electric powertrain, wherein each component of the set ofcomponents is a constraint on minimum/maximum motor torque limits forthe electric motors thereby collectively forming a set of constraints,performing, by the controller, a linear optimization to find the bestengine torque values that are attainable subject to a set of constraintinequalities as defined by current vehicle state and as monitored by theset of sensors, and controlling, by the controller, the hybrid-electricpowertrain based on the best engine torque values to avoid excessivetorque commands that could damage physical components of thehybrid-electric powertrain and/or could cause undesirablenoise/vibration/harshness (NVH) characteristics.

In some implementations, there is a plurality of constraints dependingon a configuration of the hybrid-electric powertrain and operation mode.In some implementations, the plurality of constraints includes at leasttwo of engine torque, drive belt slip torque, torque of the electricmotors, clutch and/or transmission gear states, and a state of thebattery system. In some implementations, the operation mode includes atleast one of whether the engine participates in a series hybridoperating mode or a parallel hybrid operating mode, electric motorgenerator or motor mode, battery system depletion, charge sustaining, orrecharging mode, clutch pack slippage, disengaged/open, or lockoperation, dog clutch engaged or disengaged operation, and transmissiongear selection.

In some implementations, the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations, and then the best enginetorque levels are chosen. In some implementations, the controller firstdetermines four intersection lines between the two pairs of constraintplanes and then finds the intersection points of these lines with theplanes of the motor minimum/maximum torque levels with athree-dimensional geometric approach.

In some implementations, the controller determines the intersectionpoints of two of the constraint plane pairs with the slab, and then thebest engine torque levels are chosen. In some implementations, thecontroller is configured to first find the four vertical edges of themotor minimum/maximum torque slab and then find the intersection pointsof two constraint plane pairs at a time with the four vertical edgesleading to at most sixteen intersection points.

In some implementations, the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations and determine the intersectionpoints of two of the constraint plane pairs with the slab, and then thebest engine torque levels are chosen. In some implementations, thecontroller determines four intersection lines between the two pairs ofconstraint planes and then finds the intersection points of these lineswith the planes of the motor minimum/maximum torque levels with athree-dimensional geometric approach, and finds the four vertical edgesof the motor minimum/maximum torque slab and then finds the intersectionpoints of two constraint plane pairs at a time with the four verticaledges leading to at most sixteen intersection points.

Further areas of applicability of the teachings of the presentapplication will become apparent from the detailed description, claimsand the drawings provided hereinafter, wherein like reference numeralsrefer to like features throughout the several views of the drawings. Itshould be understood that the detailed description, including disclosedembodiments and drawings referenced therein, are merely exemplary innature intended for purposes of illustration only and are not intendedto limit the scope of the present disclosure, its application or uses.Thus, variations that do not depart from the gist of the presentapplication are intended to be within the scope of the presentapplication.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram of a vehicle having an examplehybrid-electric powertrain according to the principles of the presentapplication;

FIG. 2 is a functional block diagram of an example minimum/maximumtorque limit determination architecture according to the principles ofthe present application;

FIG. 3 is an example three-dimensional graph illustratinghybrid-electric powertrain system constraints according to theprinciples of the present application;

FIG. 4 is an example three-dimensional graph illustrating pairs ofminimum/maximum linear inequality constraints according to theprinciples of the present application;

FIG. 5 is an example three-dimensional graph illustrating fourintersecting lines of two constraint plane pairs according to theprinciples of the present application;

FIG. 6 is an example three-dimensional graph illustrating four verticaledges intersecting with at most four constraint planes according to theprinciples of the present application; and

FIG. 7 is a flow diagram of an example method of determiningminimum/maximum motor torque limits based on linear constraintsaccording to the principles of the present application.

DESCRIPTION

As discussed above, a specific hybrid-electric powertrain configurationhaving an internal combustion engine, two electric motors, and atransmission comprising a plurality of clutches. The primary challengein working with a set of constraints that depend on one or both motortorque levels and the engine torque level is that they lead to morecomplex constraint equations to solve, e.g., due to different powertrainoperation modes: series or parallel hybrid operating modes, electricmotor generator or motor mode, battery system charging, discharging, orcharge sustaining mode, and clutch pack slipping, locked or disengagedoperation, dog clutch engaged or disengaged operation, transmission gearselection. Some constraint equations could be so complex that existingprocessors could lack the speed and/or memory to solve them within theupdate frequency required, which could result in the requirement of amore expensive and more powerful processor. As a result, improvedcontrol systems and methods for this configuration of hybrid-electricpowertrain are presented. These control systems and methods are capableof solving three-dimensional constraint equations in a unique and novelmanner that does not require additional processor speed/memory.

Referring now to FIG. 1 , a functional block diagram of a vehicle 100having an example hybrid-electric powertrain 104 according to theprinciples of the present application is illustrated. Thehybrid-electric powertrain 104 is configured to generate drive torquefor vehicle propulsion or regenerative braking torque at a driveline108. The hybrid-electric powertrain 104 generally comprises an optionaltransmission 112 comprising a plurality of clutches 116, an internalcombustion engine 120, two electric motors 124 a, 124 b, a batterysystem 128, and a set of one or more sensors 132. Non-limiting examplesof the sensor(s) include clutch hydraulic pressure sensor(s),transmission gear state sensor(s), shaft speed sensors, and batterysystem state sensor(s). While the electric motors 124 a, 124 b are shownseparately from the transmission 112, it will be appreciated that thetransmission 112 could be a hybrid transmission that includes one orboth of the electric motors 124 a, 124 b. A controller 136 controlsoperation of the hybrid-electric powertrain 104, such as to achieve orgenerate a desired amount of drive torque or regenerative braking torquebased on driver input or torque request via a driver interface 140.

Referring now to FIG. 2 , a functional block diagram of an exampleminimum/maximum engine torque level determination architecture 200according to the principles of the present application is illustrated.The torques of the electric motors 124 a, 124 b (also referred to as“motor A” and “motor B,” respectively) and the engine 120 are consideredto be independent variables when describing the hybrid-electricpowertrain 104: T_(a)=torque of motor A 124 a, T_(b)=torque of motor B124 b, and T_(i)=torque of engine 120 (e.g., all in Newton-meters, orNm). The mathematical modeling of some of the hybrid-electric powertraincomponents becomes more straightforward and uniform across differentvehicle powertrain architectures if a coordinate transformation isapplied to motor A and motor B torques according to the followingequations:

Tx=A ₁ *T _(a) +A ₂  (1)

Ty=B ₁ *T _(b) +B ₂  (2),

where A₁, B₁, A₂, and B₂ are known scalars, Tx (√W) representstransformed motor A torque, and Ty (√W) represents transformed motor Btorque. This coordinate transformation, however, is not a key aspect ofthe present application. A linear objective function Tm₁ is described inthe following equation:

f _(objective)(Tx,Ty,Ti)=Tm ₁ =k _(Tx,o) *Tx+k _(Ty,o) *Ty+k _(Ti,o)*Ti+D  (3)

where Tm₁ is to be optimized, k_(Tx,o), k_(Ty,o), and K_(Ti,o) are knownscalar coefficients, and D is a known scalar constant. Optimization inthis context means finding the minimum or maximum (or both) values ofTm₁ while honoring the constraint inequalities. The objective functionTm₁ can be powertrain output torque or the angular acceleration of somecomponent in the powertrain, and so on, depending on the drivingsituation and powertrain architecture.

The k_(Tx,o), k_(Ty,o), K_(Ti,o) and D values are calculated based ondifferential equations of motion of the hybrid-electric powertrain 104written out for a specific operation mode, clutch configuration, andgear configuration. Optimization in this context means finding theminimum or maximum value of Tm₁ as a function of Tx, Ty, and Ti. Thesearch for the minimum or maximum value of Tm₁ starts within aslab-shaped domain (see FIG. 3 ) determined by other software routinesupstream from these routines of the present application. Threeminimum/maximum inequality constraints forming the slab inthree-dimensions are defined by Tx (motor A torque) and Ty (motor Btorque) and Ti engine torque minimum/maximum inequality constraints asdescribed below. These minimum/maximum inequality constraintsincorporate motor torque capabilities.

Tx _(min) ≤T≤TX _(max)  (4)

Ty _(min) ≤Ty≤Ty _(max)  (5)

Ti _(min) ≤Ti≤Ti _(max)  (6).

After the Tx_(min), Tx_(max), Ty_(min), Ty_(max), Ti_(min), and Ti_(max)limits of the slab are calculated, a plurality of, for instance fivelinear minimum/maximum inequality constraints are imposed. These aredenoted Tm₂ through Tm₆.

Each of the Tm₂ through Tm₆ constraints are described generally in thefollowing pair of inequalities where n=2, . . . , 6.

Tm _(min,n) ≤k _(Tx,n) *Tx+k _(Ty,n) *Ty+k _(Ti,n) *Ti+Mn≤Tm_(max,n)  (7).

The following scalar coefficients k_(Tx,n), k_(Ty,n), and k_(Ti,n) andconstant scalar term M_(n) are determined based on the differentialequations of motion of the powertrain and some powertrain parameters,such as damping, inertia, and gear ratio values. Tm_(min,n) andTm_(max,n) are known minimum/maximum values for the n^(th) constraint.Additionally, the n^(th) minimum inequality constraint plane is parallelto the n^(th) maximum inequality constraint plane and theminimum/maximum constraints are imposed together as minimum/maximumsets. The Tm₂ through Tm₆ linear constraints each limit the spaceavailable for the minimum or maximum point. From a geometric standpoint,these constraint planes (see FIGS. 3-4 ) each may cut across the slaband leave a smaller domain available for searching for the maximum orminimum point. Descending priority is assigned to each minimum/maximumconstraint, with constraint Tm₂ having the highest priority andconstraint Tm₆ having the lowest priority.

If two or more constraint inequalities contradict one another, then theconstraint inequality with the higher priority will be considered. Morespecifically, if, for instance, a domain (e.g., domain 3) is alreadyconstrained by Tm₂ and Tm₃, and no part of domain 3 satisfies Tm₄, then,within domain 3, the geometric feature that falls the closest to the Tm₄planes will be chosen as the result of applying Tm₄. If the planes ofTm₄ are parallel with the closest plane of domain 3, then that plane ofdomain 3 will be selected as domain 4. If domain 3 has a straight edgethat runs parallel with the Tm₄ planes, and is the closest domain 3geometric feature to Tm₄, then that straight edge will be selected asdomain 4. If the domain 3 feature falling the closest to the planes ofTm₄ is a point, then that point will become domain 4. Imposing the Tm₂through Tm₆ constraints on the Tx_(min), Ty_(max), Ty_(min), Ty_(max),Ti_(min), and Ti_(max) slab and finding the Ti_(min) and Ti_(max) valuesunder those constraints is a key objective of the present application.

A summary of the hybrid-electric powertrain constraints can be seen inTable 1 below:

TABLE 1 Constraint Constraint Constraint Description Type Expression TxMin Slab, Tx min/max set Tx_(min) ≤ Tx Tx Max Slab, Tx min/max set Tx ≤Txmax Ty Min Slab, Ty min/max set Ty_(min) ≤ Ty Ty Max Slab, Ty min/maxset Ty ≤ Ty_(max) Ti Min Slab, Ti min/max set Ti_(min) ≤ Ti Ti Max Slab,Ti min/max set Ti ≤ Ti_(max) n = 2, Min Tm₂ min plane, Tm_(min, 2) ≤k_(Tx, 2)*Tx + part of the Tm₂ min, k_(Ty, 2)*Ty + k_(Ti, 2)*Ti + M₂ Tm₂max plane pair n = 2, Max Tm₂ max plane, k_(Tx, 2)*Tx + k_(Ty, 2)*Ty +part of the Tm₂ min, k_(Ti, 2)*Ti + M₂ ≤ Tm_(max, 2) Tm₂ max plane pairn = 3, Min Tm₃ min plane, Tm_(min, 3) ≤ k_(Tx, 3)*Tx + part of the Tm₃min, k_(Ty, 3)*Ty + k_(Ti, 3)*Ti + M₃ Tm₃max plane pair n = 3, Max Tm₃max plane, k_(Tx, 3)*Tx + k_(Ty, 3)*Ty + part of the Tm₃ min,k_(Ti, 3)*Ti + M₃ ≤ Tm_(max, 3) Tm₃ max plane pair n = 4, Min Tm₄ minplane, Tm_(min, 4) ≤ k_(Tx, 4)*Tx + part of the Tm₄ min, k_(Ty, 4)*Ty +k_(Ti, 4)*Ti + M₄ Tm₄max plane pair n = 4, Max Tm₄ max plane,k_(Tx, 4)*Tx + k_(Ty, 4)*Ty + part of the Tm₄ min, k_(Ti, 4)*Ti + M₄ ≤Tm_(max, 4) Tm₄ max plane pair n = 5, Min Tm₅ min plane, Tm_(min, 5) ≤k_(Tx, 5)*Tx + part of the Tm₅ min, k_(Ty, 5)*Ty + k_(Ti, 5)*Ti + M₅ Tm₅max plane pair n = 5, Max Tm₅ max plane, k_(Tx, 5)*Tx + k_(Ty, 5)*Ty +part of the Tm₅ min, k_(Ti, 5)*Ti + M₂ ≤ Tm_(max, 5) Tm₅ max plane pairn = 6, Min Tm₆ min plane, Tm_(min, 6) ≤ k_(Tx, 6)*Tx + part of the Tm₆min, k_(Ty, 6)*Ty + k_(Ti, 6)*Ti + M₆ Tm₆ max plane pair n = 6, Max Tm₆max plane, k_(Tx, 6)*Tx + k_(Ty, 6)*Ty + part of the Tm₆ min,k_(Ti, 6)*Ti + M₆ ≤ Tm_(max, 6) Tm₆ max plane pair

The hybrid-electric powertrain constraints described in Table 1 and thelinear objective function Tm₁ give rise to a three-dimensionaloptimization problem. As shown in FIG. 3 , this includes hybrid-electricpowertrain minimum/maximum linear inequality constraint planes, Tx andTy electric motor torque inequality constraints, and Ti engine torqueminimum/maximum constraints. In FIG. 3 , the hatched,parallelogram-shaped cross-section between the Tm_(i), Tm_(j) constraintplanes indicates the overlap of the Tm_(i), Tm_(j) constraints. This isan infinitely long bar that has the indicated parallelogram-shapedcross-section. The overlap bar extends behind the plane of FIG. 3 andalso protrudes out of it, towards the viewer. The [Tx, Ty, Ti] operatingpoints inside the bar satisfy both the Tm_(i) and the Tm_(j) constraintpairs, a total of four constraint planes. The determination of a Tx andTy value that satisfies the above-described hybrid-electric powertrainconstraints is performed by an already existing software routine,outside of the scope of the present application. The present applicationis instead systems and methods to determine a Ti value that satisfiesall above-described hybrid-electric powertrain constraints and achieveits minimum or maximum value within those constraints.

As discussed in some detail above, the above-described three-dimensionaloptimization problem is characterized by a plurality of minimum/maximumlinear inequality constraints (equation 7), electric motor torque Tx andTy minimum/maximum inequality constraints (equations 4-5), and enginetorque Ti minimum/maximum inequality constraints (equation 6). Theproposed technique used to determine a satisfactory minimum/maximumengine torque Ti is executed by analyzing pairs of minimum/maximumlinear inequality constraints giving consideration to priority aspreviously described herein and as shown in FIG. 4 . The minimum/maximumlinear inequality constraint of a higher priority is referred to asconstraint Tm₁. The hybrid-electric powertrain 104 shown in FIG. 1 andspecifically described herein gives rise to the novel problem ofanalyzing the following pairs of minimum/maximum linear inequalityconstraints in three dimensions: (1) one constraint depends on Tx and Tiand the other depends on Ty and Ti, (2) one constraint depends on Tx,Ty, and Ti and the other depends on Tx and Ti, (3) one constraintdepends on Tx, Ty, and Ti and the other depends on Ty and Ti, and (4)both constraints depend on Tx, Ty, and Ti. Scenarios when bothconstraints depend only on Tx and on Ti or when both constraints dependonly on Ty and on Ti are managed by already-existing software routines.Those routines are outside of the scope of the present application.Scenarios when only one or no constraint depends on Ti are also handledin other software routines that are also outside of the scope of thepresent application. According to linear programming theory, on a domainconstrained by linear constraints, a linear objective function willreach its minimum or maximum value in a point where at least threeconstraint planes intersect. The proposed technique for finding thispoint will now be discussed in greater detail. While there are amultitude of, for instance five constraint plane pairs, they will beanalyzed only two at a time, thereby saving on processor throughput andmemory. The Tm and Tm₁ constraints are selected in block 208 of FIG. 2from a set of constraints determined at block 204. Each possiblecombinations of constraint plane pairs are selected and analyzed. Theorder of execution of these pairwise analyses is irrelevant. After allpairs are analyzed in block 224 (comparison of at most 24 Ti values forminimum/maximum), the results will be synthesized in block 208 (alsoreferred to as “running arbitration”) and the final results for TiMinand TiMax are sent out in block 228 for use in controlling thehybrid-electric powertrain 204.

According to the proposed technique, first the TiMin, TiMax constraintis removed. From here on, it is assumed that the Tx, Ty, Ti constraintslab is infinitely tall and deep in the Ti direction. On this extendeddomain, all intersection points will be identified and logged. There aretwo types of intersection points. For the first type of intersectionpoint, the location of the points where the constraint planeintersection lines poke into and out of the constraint slab isdetermined at blocks 212 and 216 of FIG. 2 . The two pairs of constraintplanes have up to four intersection lines, which can be seen in FIG. 5 .The parameters of the intersection lines are determined in block 212.The points where each of the four intersection lines enter and exit theslab will be located. Each of the four lines can have at most two suchpoints on it, so this can yield a maximum of eight points, which arefound in block 216. This process will now be described in greaterdetail.

These points can lie along the line formed by the intersection of twoconstraint planes or intersections of constraint planes and the slabformed by the Tx, Ty, and Ti inequality constraints. First, the fourlines formed by intersections of the constraint planes must be computed.The following equations describe the four constraint planes:

k _(Tx,i)(Tx)+K _(Ty,i)(Ty)+K _(Ti,i)(Ti)+M _(i) ≥Tm _(min,i)

k _(Tx,j)(Tx)+K _(Ty,j)(Ty)+K _(Ti,j)(Ti)+M _(j) ≥Tm _(min,j)

k _(Tx,i)(Tx)+K _(Ty,i)(Ty)+K _(Ti,i)(Ti)+M _(i) ≤Tm _(max,i)

k _(Tx,j)(Tx)+K _(Ty,j)(Ty)+K _(Ti,j)(Ti)+M _(j) ≤Tm _(max,j.)

The M_(i) and M_(j) variables are scalar constants, and so are theTm_(min,i), Tm_(min,j), Tm_(max,i), and Tm_(max,j) values. Thus, scalarconstants Q_(min,i), Q_(min,j), Q_(max,i), and Q_(max,j) can be definedas follows:

Q _(min,i) =Tm _(min,i) −M _(i)

Q _(min,j) =Tm _(min,j) −M _(j)

Q _(max,i) =Tm _(max,i) −M _(i)

Q _(max,j) =Tm _(max,j) −M _(j).

Thus, the four constraint planes can be expressed generally as thefollowing intermediate equations:

k _(Tx,i)(Tx)+K _(Ty,i)(Ty)+K _(Ti,i)(Ti)≥Q _(min,i)  (8a)

k _(Tx,j)(Tx)+K _(Ty,j)(Ty)+K _(Ti,j)(Ti)≥Q _(min,j)  (8b)

k _(Tx,i)(Tx)+K _(Ty,i)(Ty)+K _(Ti,i)(Ti)≤Q _(max,i)  (9a)

k _(Tx,j)(Ty)+K _(Ty,j)(Ty)+K _(Ti,j)(Ti)≤Q _(max,i).  (9b)

Since minimum inequality constraint plane i is parallel to the maximuminequality constraint plane i and minimum inequality constraint plane jis parallel to the maximum inequality constraint plane j, intersectionsare only found between the i and j planes and not the i and i or j and jplanes. The procedure described below provides a technique to find oneintersection line and is eventually expanded to four intersection lines(see FIGS. 4-5 ).

A line formed by the intersection of two linear inequality constraintplanes is expressed generally in the following equation:

{right arrow over (r)}(λ)=P ₀ +λ{right arrow over (d)}  (10)

where λ is an introduced scalar parameter, P₀=

Tx₀,Ty₀,Ti₀

is a point that lies on both linear inequality constraint planes, and{right arrow over (d)}={right arrow over (N)}_(Tm,i)×{right arrow over(N)}_(Tm,j) is the direction vector of the line of intersection, {rightarrow over (N)}_(Tm,i)=

k_(Tx,i),k_(Ty,i),k_(Ti,i)

and {right arrow over (N)}_(Tm,j)=

k_(Tx,j),k_(Ty,j),k_(Ti,j)

are the normal vectors to the linear inequality constraint planes. Thecross-product of the two normal vectors is the direction vector of theintersection line. Because there are two pairs of parallel planes, thefour direction vectors of the four lines will be parallel. It issufficient to calculate only one of them. P₀ is an arbitrary point alongthe line. To find P₀, consider that the intersection line will cross atleast two of the following three planes: Tx=0, Ty=0, and Ti=0. P₀ can befound by setting one of the three variables (Tx, Ty, or Ti) to be zeroand solving the system of two linear equations for the other tworemaining variables. The selection of which of the three variables thatis set to zero depends on the direction of the intersection line. If theline is known to intersect the Tx=0 plane, then solve the equations fora P₀ point on the Tx=0 plane. If the line is known to intersect the Ty=0plane, then solve the equations for Ty=0. If the line is known tointersect the Ti=0 plane, then solve the equations for Ti=0. Every lineintersects a combination of two or all three of these planes, so inevery case it will be possible to choose one plane and solve for P₀ onthat plane.

In the following equations, for instance, Ty₀=0 was chosen. To find apoint P_(0,i_min,j_max) along the intersection line of the Tm_(i,min)and Tm_(j,max) constraint planes, the following equations must besolved:

$\begin{matrix}{{\begin{bmatrix}k_{{Tx},i} & k_{{Ti},i} \\k_{{Tx},j} & k_{{Ti},j}\end{bmatrix}\begin{bmatrix}{Tx}_{{@{Ty}} = 0} \\{Ti}_{{@{Ty}} = 0}\end{bmatrix}} = \begin{bmatrix}Q_{i,\min} \\Q_{j,\max}\end{bmatrix}} & ( {11a} )\end{matrix}$ $\begin{matrix}{\begin{bmatrix}{Tx}_{{@{Ty}} = 0} \\{Ti}_{{@{Ty}} = 0}\end{bmatrix} = {\begin{bmatrix}k_{{Tx},i} & k_{{Ti},i} \\k_{{Tx},j} & k_{{Ti},j}\end{bmatrix}^{- 1}\begin{bmatrix}Q_{i,\min} \\Q_{j,\max}\end{bmatrix}}} & ( {11b} )\end{matrix}$ $\begin{matrix}{{{Tx}_{0} = \frac{{( Q_{i,\min} )( k_{{Ti},j} )} - {( Q_{j,\max} )( k_{{Ti},i} )}}{{( k_{{Tx},i} )( k_{{Ti},j} )} - {( k_{{Ti},i} )( k_{{Tx},j} )}}}{{Ty}_{0} = 0}{{Ti}_{0} = \frac{{{- ( Q_{i,\min} )}( k_{{Tx},j} )} + {( Q_{j,\max} )( k_{{Tx},i} )}}{{( k_{{Tx},i} )( k_{{Ti},j} )} - {( k_{{Ti},i} )( k_{{Tx},j} )}}}} & (12)\end{matrix}$

The equations above determine the coordinates Tx₀, Ty₀, Ti₀ of the pointP_(0,i_min,j_max). The coordinates of the point P_(0,i_min,j_min) alongthe intersection line of Tm₁,min and Tm₁,min and the pointP_(0,i_max,j_max) along the intersection line of Tm_(i_max) andTm_(j_max) and P_(0,i_max,j_min) along the intersection of Tm_(i_max)and Tm_(j_min) can be calculated similarly. These procedures take placein block 212 of FIG. 2 . An analogous procedure can be followed if thestarting assumption is Tx=0 or Ti=0.

Generally, the line of intersection between two minimum/maximum linearinequality constraint planes is shown in the following equation:

{right arrow over (r)}(λ)=P ₀ +λ{right arrow over (d)}=

Tx ₀ +λd _(Tx) ,Ty ₀ +λd _(Ty) ,Ti ₀ +λd _(Ti)

  (13)

where Ty₀=0 and the parameter λ can be found by setting the Tx and Tycomponents of the line described in the equations below equal to theextreme values of the inequality constraints described in equations 4-6that make up the slab in three dimensions. These A values willeventually be used to compute Ti at these points, as shown in theequations below:

$\begin{matrix}{{{Tx}{Min}} = {{Tx}_{0} + {\lambda_{TxMin} \cdot d_{Tx}}}} & ( {14a} )\end{matrix}$ $\begin{matrix}{{\lambda_{TxMin} = \frac{{{Tx}{Min}} - {Tx}_{0}}{d_{Tx}}}{{{Tx}{Max}} = {{Tx}_{0} + {\lambda_{TxMax} \cdot d_{Tx}}}}{\lambda_{TxMax} = \frac{{{Tx}{Max}} - {Tx}_{0}}{d_{Tx}}}} & ( {14b} )\end{matrix}$

where λ_(TxMin) corresponds to the point where one of the fourconstraint intersection lines pokes through the Tx=Tx_(Min) plane on theconstraint slab and λ_(TxMax) corresponds to the point where one of thefour constraint intersection lines pokes through the Tx=Tx_(Max) planeon the constraint slab. Continuing the same logic for the remaining twoplanes of the slab:

TyMin=Ty ₀+λ_(TyMin) ·d _(Ty)

P₀ can be solved for on the Ty=0 plane in the present example. Ty₀=0 isthe Ty coordinate of the P₀ point used to create the equation of thisintersection line, so:

$\begin{matrix}{{{{Ty}{Min}} = {\lambda_{TyMin} \cdot d_{Ty}}}{\lambda_{TyMin} = {\frac{{Ty}{Min}}{d_{Ty}}.}}} & ( {14c} )\end{matrix}$

Similarly:

$\begin{matrix}{\lambda_{TyMax} = {\frac{{Ty}{Max}}{d_{Ty}}.}} & ( {14d} )\end{matrix}$

After the λ_(TxMin), λ_(TxMax), λ_(TyMin), and λ_(TyMax) values arefound, equation 13 is used to determine the four Ti values for one ofthe intersection lines. After finding the points, each must be checkedagainst the slab constraints described in equations 4-6. The followinginequality verifies if an intersection point on the Ty=Ty_(Min) planesatisfies the Tx_(Min) and Tx_(Max) constraints:

TxMin≤Tx ₀+λ_(TyMin) ·d _(Tx) ≤TxMax  (15a).

Similarly, the following inequality verifies if an intersection point onthe Ty=Ty_(Max) plane satisfies the Tx_(Min) and Tx_(Max) constraints:

TxMin≤Tx ₀+λ_(TyMax) ·d _(Tx) ≤TxMax  (15b).

Similarly, for the Ty_(Min)<Ty and the Ty<Ty_(Max) constraints, with Ty₀still being set to zero:

TyMin≤λ_(Txmin) ·d _(Ty) ≤TyMax  (15c)

TyMin≤λ_(TxMax) ·d _(Ty) ≤TyMax  (15d).

Only those points are kept for further analysis that satisfy theconstraints. As done in the following equations, Ti is calculated usingeach A that satisfies the above constraints imposed by the slabdescribed in equations 4-6:

Ti ₁ =Ti ₀+λ₁ d _(Ti)  (16a)

Ti ₂ =Ti ₀+λ₂ d _(Ti)  (16b).

The above procedure is executed for intersection lines 1, 2, 3, and 4formed by the pairs of Tm_(i), Tm_(j) constraint plane pairs, leading toa total of at most 8 Ti values. This procedure is carried out in block216 of FIG. 2 .

For the second type of intersection point, the points where the verticaledges of the slab intersect with the four constraint planes are locatedin block 220 of FIG. 2 . The four edges of the Tx, Ty, Ti limit slabform the four contour lines that run parallel with the Ti axis and aredefined as “vertical edges” and can be seen in FIG. 6 . These four edgesmay each poke through some or all of the four constraint planes, leadingto a total of at most 16 intersection points. For instance, “VerticalEdge 2” is at Tx=Tx_(Max), Ty=Ty_(Min). To obtain the intersection pointof “Vertical Edge 2” with the Tm_(j,max) plane, the coordinates:Tx=TxMax and Ty=TyMin are substituted into the equation of theTm_(j,max) plane. The equation of the Tm_(j,max) plane is rearranged tocalculate the Ti coordinate of the intersection point. After thesepoints are determined, they are individually checked if they fall withinthe hatched overlap volume of the Tm and Tm₁ constraints, as shown inFIG. 4 . Those points that do are kept for further processing. Afterfinding these at most 8+16=24 points, the points with the minimum andmaximum Ti values are selected in blocks 224 and 208 as described ingreater detail below.

If the current TiMin is smaller than the previously stored TiMin, thenthis current TiMin is kept. Otherwise, the previously stored TiMin iskept. Similarly, if the current TiMax is larger than the previouslystored TiMax, then the current TiMax is kept. Otherwise, we keep thepreviously stored TiMax. As described in greater detail below, thisprocedure is carried out for all possible combinations of two of themultiplicity of constraint plane pairs, and then the minimum and maximumTi values are chosen from all the calculated values. This repetitiveprocess runs around the feedback loop labeled “TiMin & TiMax fromconstraints” in FIG. 2 .

Referring now to FIG. 7 , a flow diagram of an example method 700 ofdetermining minimum and maximum engine torque limits based onthree-dimensional constraints and controlling a hybrid-electricpowertrain according to the principles of the present application isillustrated. While the components of the hybrid-electric powertrain 104are specifically referenced for illustrative purposes, it will beappreciated that the method 700 could be applicable to other suitablehybrid-electric powertrain configurations. At 704, the controller 136 ofthe vehicle 100 and from a set of sensors 132 of the hybrid-electricpowertrain 104, a set of operating parameters of a set of components ofthe hybrid-electric powertrain 104, wherein each component of the set ofcomponents may impose a constraint on the motor and engine torque levelsfor the electric motors 124 a, 124 b and engine 120. At 708, thealgorithm shown in FIG. 2 is executed. At 712, the controller 136accesses a three-dimensional graph including a slab representing variousminimum/maximum motor torque limits for the engine 120 and the electricmotors 124 a, 124 b. At 712, based on the engine torque limitsdetermined at 708 are further processed and the torque command signalsto the motors and the engine are sent out. At 712, the controller 136controls the hybrid-electric powertrain 104 based on the final minimumand maximum engine torque values to avoid excessive torque commands thatcould damage physical components of the hybrid-electric powertrainand/or could cause undesirable noise/vibration/harshness (NVH)characteristics. The method 700 then ends or returns to 704 for one ormore additional cycles.

It should also be understood that the mixing and matching of features,elements, methodologies and/or functions between various examples may beexpressly contemplated herein so that one skilled in the art wouldappreciate from the present teachings that features, elements and/orfunctions of one example may be incorporated into another example asappropriate, unless described otherwise above.

What is claimed is:
 1. A control system for a hybrid-electric powertrainof a vehicle, the hybrid-electric powertrain including an internalcombustion engine, two electric motors, a battery system, and atransmission comprising a plurality of clutches, the control systemcomprising: a set of sensors configured to measure a set of operatingparameters of a set of components of the hybrid-electric powertrain,wherein each component of the set of components is a constraint onminimum/maximum motor torque limits for the electric motors therebycollectively forming a set of constraints; and a controller configuredto: perform a linear optimization to find the best engine torque valuesthat are attainable subject to a set of constraint inequalities asdefined by current vehicle state and as monitored by the set of sensors;and control the hybrid-electric powertrain based on the best enginetorque values to avoid excessive torque commands that could damagephysical components of the hybrid-electric powertrain and/or could causeundesirable noise/vibration/harshness (NVH) characteristics.
 2. Thecontrol system of claim 1, wherein there are a plurality of constraintsdepending on a configuration of the hybrid-electric powertrain andoperation mode.
 3. The control system of claim 2, wherein the pluralityof constraints includes at least two of engine torque, drive belt sliptorque, torque of the electric motors, clutch and/or transmission gearstates, and a state of the battery system.
 4. The control system ofclaim 2, wherein the operation mode includes at least one of whether theengine participates in a series hybrid operating mode or a parallelhybrid operating mode, electric motor generator or motor mode, batterysystem depletion, charge sustaining, or recharging modes, clutch packslippage, disengaged/open, or lock operation, dog clutch engaged ordisengaged operation, and transmission gear selection.
 5. The controlsystem of claim 1, wherein the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations, and then the best enginetorque levels are chosen.
 6. The control system of claim 5, wherein thecontroller first determines four intersection lines between the twopairs of constraint planes and then finds the intersection points ofthese lines with the planes of the motor minimum/maximum torque levelswith a three-dimensional geometric approach.
 7. The control system ofclaim 1, wherein the controller determines the intersection points oftwo of the constraint plane pairs with the slab, and then the bestengine torque levels are chosen.
 8. The control system of claim 7,wherein the controller is configured to first find the four verticaledges of the motor minimum/maximum torque slab and then find theintersection points of two constraint plane pairs at a time with thefour vertical edges leading to at most sixteen intersection points. 9.The control system of claim 1, wherein the controller is configured totake two constraint plane pairs at a time and determine theintersections between them and the motor torque minimum/maximum limitsfor all currently active constraint plane pair combinations anddetermine the intersection points of two of the constraint plane pairswith the slab, and then the best engine torque levels are chosen. 10.The control system of claim 9, wherein the controller determines fourintersection lines between the two pairs of constraint planes and thenfinds the intersection points of these lines with the planes of themotor minimum/maximum torque levels with a three-dimensional geometricapproach, and finds the four vertical edges of the motor minimum/maximumtorque slab and then finds the intersection points of two constraintplane pairs at a time with the four vertical edges leading to at mostsixteen intersection points.
 11. A method for controlling for ahybrid-electric powertrain of a vehicle, the hybrid-electric powertrainincluding an internal combustion engine, two electric motors, a batterysystem, and a transmission comprising a plurality of clutches, themethod comprising: receiving, by a controller of the vehicle and from aset of sensors of the hybrid-electric powertrain, a set of operatingparameters of a set of components of the hybrid-electric powertrain,wherein each component of the set of components is a constraint onminimum/maximum motor torque limits for the electric motors therebycollectively forming a set of constraints; performing, by thecontroller, a linear optimization to find the best engine torque valuesthat are attainable subject to a set of constraint inequalities asdefined by current vehicle state and as monitored by the set of sensors;and controlling, by the controller, the hybrid-electric powertrain basedon the best engine torque values to avoid excessive torque commands thatcould damage physical components of the hybrid-electric powertrainand/or could cause undesirable noise/vibration/harshness (NVH)characteristics.
 12. The method of claim 11, wherein there are aplurality of constraints depending on a configuration of thehybrid-electric powertrain and operation mode.
 13. The method of claim12, wherein the plurality of constraints includes at least two of enginetorque, drive belt slip torque, torque of the electric motors, clutchand/or transmission gear states, and a state of the battery system. 14.The method of claim 12, wherein the operation mode includes at least oneof whether the engine participates in a series hybrid operating mode ora parallel hybrid operating mode, electric motor generator or motormode, battery system depletion, charge sustaining, or recharging mode,clutch pack slippage, disengaged/open, or lock operation, dog clutchengaged or disengaged operation, and transmission gear selection. 15.The method of claim 11, wherein the controller is configured to take twoconstraint plane pairs at a time and determine the intersections betweenthem and the motor torque minimum/maximum limits for all currentlyactive constraint plane pair combinations, and then the best enginetorque levels are chosen.
 16. The method of claim 15, wherein thecontroller first determines four intersection lines between the twopairs of constraint planes and then finds the intersection points ofthese lines with the planes of the motor minimum/maximum torque levelswith a three-dimensional geometric approach.
 17. The method of claim 11,wherein the controller determines the intersection points of two of theconstraint plane pairs with the slab, and then the best engine torquelevels are chosen.
 18. The method of claim 17, wherein the controller isconfigured to first find the four vertical edges of the motorminimum/maximum torque slab and then find the intersection points of twoconstraint plane pairs at a time with the four vertical edges leading toat most sixteen intersection points.
 19. The method of claim 11, whereinthe controller is configured to take two constraint plane pairs at atime and determine the intersections between them and the motor torqueminimum/maximum limits for all currently active constraint plane paircombinations and determine the intersection points of two of theconstraint plane pairs with the slab, and then the best engine torquelevels are chosen.
 20. The method of claim 19, wherein the controllerdetermines four intersection lines between the two pairs of constraintplanes and then finds the intersection points of these lines with theplanes of the motor minimum/maximum torque levels with athree-dimensional geometric approach, and finds the four vertical edgesof the motor minimum/maximum torque slab and then finds the intersectionpoints of two constraint plane pairs at a time with the four verticaledges leading to at most sixteen intersection points.